Setting aside the actions of commuting diagonal flows, a next major step for actions on quotients \(X\)  by lattices \(\Lambda <G\)  is reached by the seminal work of Benoist–Quint [1]}, [2]}, [3]}, [4]}. Their work involved describing dynamics of actions by subgroups \(\Gamma \)  whose algebraic (Zariski) closure has semisimplicity but the subgroups themselves can be genuinely irregular, e.g.  discrete. Focusing on stationary measures of random walks on quotients \(X\) , they developed the exponential drift argument used to obtain a description of all stationary measures. The drift argument of Benoist–Quint requires a precise control of random matrix products (e.g.  local limit theorem), a feature not readily available without a semisimplicity assumption. Benoist–Quint's ideas were then remarkably modified in a non-homogeneous setting by Eskin–Mirzakhani [5]} who managed to set up a much more flexible argument bypassing for example the need for a local limit theorem. This development enabled further extensions of Benoist–Quint's results in several directions by Eskin–Lindenstrauss [6]}, [7]}. Some of our arguments in this article (e.g.  the
six points drift argument) draws on the ideas of the latter works [5]}, [6]} and in fact can be seen as a slightly modified and simpler version of them.
